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A370721 Positive integers k == 2 (mod 4) such that the parametric Pell-type equation x^2 - m*x*y + y^2 = m^2 + k has no integer solutions (x,y) for all integer m >= 1.

%I A370721 #100 Jun 12 2024 08:34:00
%S A370721 14,94,114,118,154,158,214,238,254,294,358,414,478,574,594,598,614,
%T A370721 654,658,694,718,758,790,814,834,862,874,878,934,958,994,1014,1054,
%U A370721 1106,1174,1198,1294,1414,1434,1454,1486,1494,1498,1558,1634,1678,1738,1774,1794,1834,1894,1918,1978
%N A370721 Positive integers k == 2 (mod 4) such that the parametric Pell-type equation x^2 - m*x*y + y^2 = m^2 + k has no integer solutions (x,y) for all integer m >= 1.
%C A370721 For a positive integer k == 2 (mod 4), it suffice to check that the equation x^2-m*x*y+y^2 = m^2+k (*) has no integer solutions (x,y) for all integer m with 1 <= m <= k/2 (see references for the proof of some similar assertions). This condition can be verified by an algorithm similar to brute force search for the general Pell equation x^2-Dy^2 = N (see, for example, sect. 4.4.5 in: Andreescu T., Andrica D. Quadratic Diophantine Equations. New York: Springer, 2015).
%C A370721 Also, the equation (*) has no integer solutions (x,y) for all integer m >= 1 when k = 1 or k = 4. For any other positive integer k, the equation (*) has integer solutions (x,y) for infinitely many integers m >= 1.
%D A370721 N. Osipov, A Pell-Type Diophantine Equation, Amer. Math. Monthly, 128 (2021), p. 858-860.
%D A370721 N. Osipov, A Pell-type Equation in Disguise, Amer. Math. Monthly, 129 (2022), p. 389-390.
%H A370721 Orlov Nikita, <a href="/A370721/a370721.pas.txt">Pascal program</a>.
%p A370721 check:=proc(k) local flag,y,m,yy,mm; flag:=0;
%p A370721 for y from 0 to evalf(2*sqrt((k+1)/3)+1) while flag=0 do
%p A370721 if issqr(-3*y^2+4*k+4)=true then flag:=1; mm:=1; yy:=y; fi; od;
%p A370721 for m from 3 to k/2 while flag=0 do
%p A370721 if m mod 4<>2 then for y from 0 to evalf(sqrt((m^2+k)/(m+2)))+1 while flag=0 do
%p A370721 if issqr((m^2-4)*y^2+4*(m^2+k))=true then flag:=1; mm:=m; yy:=y; fi; od; fi; od;
%p A370721 if flag=0 then return 0 else return [mm,yy]; fi; end proc:
%p A370721 for k from 1 to 2000 do if k mod 4=2 and check(k)=0 then print(k); fi; od:
%o A370721 (Pascal) (* see link *)
%Y A370721 Cf. A371957 (for the equation x^2-m*x*y+y^2=-m^2-k).
%K A370721 nonn
%O A370721 1,1
%A A370721 _Orlov Nikita_ and _Nikolay Osipov_, Mar 07 2024
%E A370721 Edited by _Nikolay Osipov_, Jun 11 2024